Preheat your oven to 325°F (163°C). Grease and line a 9-inch round cake pan with parchment paper. In a heatproof bowl, melt the butter and dark chocolate together over a double boiler or in the microwave, stirring until smooth.
In a large bowl, whisk together the granulated sugar, brown sugar, eggs, and vanilla extract. Gradually mix in the melted chocolate mixture until well combined.
Sift together the flour, cocoa powder, baking powder, and salt. Gradually fold the dry ingredients into the wet mixture. Pour in the hot coffee and stir until smooth. The batter will be thick and glossy.
Pour the batter into the prepared cake pan and bake for 35-40 minutes, or until a toothpick inserted in the center comes out with a few moist crumbs. Let the cake cool in the pan for 10 minutes before transferring it to a wire rack.